Getting Around Taiwan
The public transit system in Taiwan is excellent. Most Taiwanese people utilize a combination of buses, trains, and the MRT (underground) when getting around in the cities. On all public transit, it's important to note that the Taiwanese are very polite and will always give up their seat to the elderly, disabled, pregnant, and women with children (and you should too!). Specifically, the first several chairs on the bus (which are sometimes a different color than the other chairs) and dark blue chairs on the MRT are reserved for these groups. If you see any of these people standing, you'd be well advised to surrender your seat to them. It's possible that someone will ask/tell you to move if you don't.
Taxis: Taxis are ubiquitous in Taiwan and ready to take you where you need to go. Some foreigners have complained of overly-long rides, but while taxi-drivers are generally unable to speak English, I’ve encountered nothing but quick, reliable service if you are clear about where you want to go. They are always metered and the drivers don't usually try to run the meter. In Taoyuan, the meter starts at 80NT. In Taipei, it's 70NT. Note that unless you know how to properly pronounce the name of a well-known landmark in Chinese, you need to have the address of your destination written in Chinese. It's a good idea to keep a copy of your own address, written in Chinese, with you at all times as well. Cost for a taxi is relatively cheap in comparison to American taxis, with a 15 minute drive costing somewhere around 120NT (4$USD). There is no tipping, so don’t hesitate to take a cab after a long night, lest you tip something else... namely, your scooter.
High-Speed Rail (HSR): There are multiple trains that run from the top to the bottom of Taiwan. The High Speed Rail runs on its own track, independent from local trains. You will likely want to take this train for longer-distance trips, vacations, etc. It runs from Taipei to Kaohsiung (Zuoying station). A ticket from Taipei to Kaohsiung is 1200NT (about $35US) and well worth it, as HSR tickets guarantee you a seat on a clean, efficiently run, fast train. A meal cart comes around selling chips, lunch boxes and drinks. Beware that the stations can sometimes be located somewhat outside the city center and you will probably need to take a taxi or courtesy bus to get into the city.
Local Trains: Many towns have a local train stop that will allow you to travel between towns and cities. The cost varies depending on the speed of train you choose, but is generally around 50 or 60NT (~3$USD). Chances are, no one will check your ticket on the slow train and you'll be standing the whole way. If you get on a faster train (with rows of chairs), you can buy a ticket for a certain seat and sit down, or use your EasyCard and stand. If someone is sitting in your seat, be sure to double check that you're in the right car and seat, but you can ask them to move.
The MRT/Easy Card: In Taipei, you can buy an Easy Card which can be used on trains, the MRT and city buses. For a small deposit (~100NT) you get a rechargeable card that you can keep adding money to as necessary. Inside the MRT station there are machines where you slip your card in a slot and feed money into a vending machine to add value. You get a 20% discount on your fare when you use the Easy Card.
One day MRT passes are available for $200NT (including a refundable 50NT deposit), as well as blue tokens for single trips. It's usually between 20 and 50NT to go from any one stop to another. Also note it is STRICTLY FORBIDDEN to eat/drink on the MRT and in any part of the MRT station. This includes water, gum and candy. You will be fined.
Buses: Buses are convenient and widely used although the stops can be confusing as they are generally in Chinese, as are the names of the buses themselves. Unless you know where you're going and where you need to get off, you're probably best sticking to the train or taxis. Change is unavailable so you either need to pay the exact amount or just not get change back. In downtown Taoyuan, you can buy a bus card at a ticketing station to the right of KFC when you're facing the train station. There's a deposit of $100NT but they're very convenient. Note that you need to beep your card when you get on AND off the bus.